nLab amnestic isofibration

Idea

A functor is an amnestic isofibration if it is an amnestic functor and an isofibration. The combination of these properties admits a particularly simply description in terms of a lifting property.

Proposition

A functor $U : \mathcal{D} \to \mathcal{C}$ is an amnestic isofibration if and only if, for any object $D$ in $\mathcal{D}$ and any isomorphism $f : C \to U D$ in $\mathcal{C}$, there is a unique isomorphism $\tilde{f} : \tilde{C} \to D$ such that $U \tilde{f} = f$.

Proof

Indeed, if $\tilde{f}' : \tilde{C}' \to D$ were any other isomorphism such that $U \tilde{f}' = f$, then $U (\tilde{f}^{-1} \circ \tilde{f}') = id_C$, so we must have $\tilde{f} = \tilde{f}'$.

Amnestic isofibrations are occasionally called discrete isofibrations, but this term may be misleading, because they are not isofibrations with discrete fibres.

Properties

Lemma

An equivalence of categories is an isomorphism if and only if it is an amnestic isofibration.

Proof

Every equivalence is essentially surjective on objects. If it is an isofibration, it is actually surjective on objects. If it is fully faithful and amnestic, then it is injective on objects. Under the given assumptions, the equivalence is thus surjective and injective on objects, and full faithfulness implies bijectivity on morphisms, hence it is an isomorphism.

Proposition

A monadic functor is strictly monadic if and only if it is also an amnestic isofibration.

Proof

Clearly, a strictly monadic functor is an amnestic isofibration; and if a monadic functor $U$ is amnestic, then the comparison functor $K$ is also amnestic, and if $U$ is a monadic isofibration, so is $K$; therefore in this case $K$ must be an isomorphism of categories.

Examples

• If $F : A \to B$ is a bijective on objects functor, then $[F, C] : [B, C] \to [A, C]$ is an amnestic isofibration (Lemma 2.4.8 of Avery 2017).

Related concepts

• amnestic functor
• isofibration
• conservative functor
• monadic functor

References

A reference using the term amnestic isofibrations:

• Tom Avery: Structure and Semantics [arXiv:1708.01050]

A reference using the term discrete isofibrations:

• Steve Lack: A 2-categories companion, in John Baez, Peter May: Towards Higher Categories, Springer, (2009) [arXiv:math/0702535]